Changes between Version 21 and Version 22 of ComplexMassScheme

Timestamp:

Aug 13, 2015, 3:41:57 AM (9 years ago)

Author:

Valentin Hirschi

Comment:

--

ComplexMassScheme

@@ -11 +11 @@
  * Because of the modified onshell renormalization condition in the CMS, the logarithms appearing in the UV wavefunction renormalization counterterms must be evaluated in the correct Riemann sheet.
 
-The details of these issues will be discussed in a forthcoming publication;  this wiki page is mainly to describe the various options to the command '{{{check cms}}}' which automatically tests the consistency of the CMS implementation. The core idea of the test is to compare amplitudes in the CMS scheme ($\mathcal{A}_{\text{CMS}}$) and in the case of widths set to zero ($\mathcal{A}_{\Gamma=0}$) for a given kinematic configuration where all resonances are far off-shell.
-The difference between these two amplitudes must be of higher order. More formally, this means that if we have $\mathcal{A}^{\text{Born}}_{\text{CMS}}\sim \mathcal{A}^{\text{Born}}_{\Gamma=0} \sim \mathcal{O}(\alpha^a)$, then we can write the following:
+The details of these issues will be discussed in a forthcoming publication;  this wiki page is mainly to describe the various options to the command '{{{check cms}}}' which automatically tests the consistency of the CMS implementation. The core idea of the test is to compare squared amplitudes in the CMS scheme ($\mathcal{M}_{\text{CMS}}$) and in the case of widths set to zero ($\mathcal{M}_{\Gamma=0}$) for a given kinematic configuration where all resonances are far off-shell.
+The difference between these two amplitudes must be of higher order. More formally, this means that if we have $\mathcal{M}^{\text{Born}}_{\text{CMS}}\sim \mathcal{M}^{\text{Born}}_{\Gamma=0} \sim \mathcal{O}(\alpha^a)$, then we can write the following:
 
-At LO, $(\mathcal{A}^{\text{Born}}_{\text{CMS}}-\mathcal{A}^{\text{Born}}_{\Gamma=0})/\alpha^a \equiv \Delta^{\text{LO}} = \kappa^{\text{LO}}_0 + \kappa^{\text{LO}}_1\alpha + \mathcal{O}(\alpha^2) $. The statement that the difference is of higher order is then equivalent to state that $\kappa^{\text{LO}}_0=0$. 
+At LO, $(\mathcal{M}^{\text{Born}}_{\text{CMS}}-\mathcal{M}^{\text{Born}}_{\Gamma=0})/\alpha^a \equiv \Delta^{\text{LO}} = \kappa^{\text{LO}}_0 + \kappa^{\text{LO}}_1\alpha + \mathcal{O}(\alpha^2) $. The statement that the difference is of higher order is then equivalent to state that $\kappa^{\text{LO}}_0=0$. 
 
 At NLO, this relation becomes
-$((\mathcal{A}^{\text{Virtual}}_{\text{CMS}}+\mathcal{A}^{\text{Born}}_{\text{CMS}})-(\mathcal{A}^{\text{Virtual}}_{\Gamma=0}+\mathcal{A}^{\text{Born}}_{\Gamma=0}))/\alpha^{a+1} \equiv \Delta^{\text{NLO}} = \kappa^{\text{NLO}}_0 + \kappa^{\text{NLO}}_1\alpha + \mathcal{O}(\alpha^2) $
+$((\mathcal{M}^{\text{Virtual}}_{\text{CMS}}+\mathcal{M}^{\text{Born}}_{\text{CMS}})-(\mathcal{M}^{\text{Virtual}}_{\Gamma=0}+\mathcal{M}^{\text{Born}}_{\Gamma=0}))/\alpha^{a+1} $ $\equiv \Delta^{\text{NLO}} = \kappa^{\text{NLO}}_0 + \kappa^{\text{NLO}}_1\alpha + \mathcal{O}(\alpha^2) $
 
 In order to check that $\kappa^{\text{LO}}_0$ and $\kappa^{\text{NLO}}_0$ are indeed zero, the test proceeds by scaling down all relevant couplings and widths by the parameter $\lambda$ and evaluate the expressions of $\Delta$ for many progressively smaller values of $\lambda$, but always on the same offshell kinematic configuration. One can then plot the quantities $\Delta^{\text{NLO|LO}}/\lambda$ and make sure that the asymptot for small values of lambda is the constant $\kappa^{\text{NLO|LO}}_1$. Any divergent behavior would be a manifestation of the presence of the term $\kappa^{\text{NLO|LO}}_0/\lambda$ which reveals an issue with the CMS implementation (most likely one of the two points mentioned above) which spoils the expected cancellation.